Fast algorithms for sparse portfolio selection considering industries and investment styles

Abstract

In this paper, we consider a large scale portfolio selection problem with and without a sparsity constraint. Neutral constraints on industries are included as well as investment styles. To develop fast algorithms for the use in the real financial market, we shall expose the special structure of the problem, whose Hessian is the summation of a diagonal matrix and a low rank modification. Specifically, an interior point algorithm taking use of the Sherman–Morrison–Woodbury formula is designed to solve the problem without any sparsity constraint. The complexity in each iteration of the proposed algorithm is shown to be linear with the problem dimension. In the occurrence of a sparsity constraint, we propose an efficient three-block alternating direction method of multipliers, whose subproblems are easy to solve. Extensive numerical experiments are conducted, which demonstrate the efficiency of the proposed algorithms compared with some state-of-the-art solvers.